MHB Evaluate the definite integral x/√(e^x+(2+x)^2)

[lfdahl]

Evaluate

$$I = \int_{-2}^{0} \frac{x}{\sqrt{e^x+(2+x)^2}}\,dx$$

 

[lfdahl]

Suggested solution:

Spoiler

Use the substitution:

\[v = (x+2)e^{-x/2}.\]

\[dv = -\frac{x}{2}e^{-x/2}dx\]

\[\frac{xdx}{\sqrt{e^x+(x+2)^2}} = \frac{xe^{-x/2}dx}{\sqrt{1+(x+2)^2e^{-x}}} = \frac{-2dv}{\sqrt{1+v^2}}\]

Hence, the integral becomes:

\[I = \int_{-2}^{0}\frac{xdx}{\sqrt{e^x+(x+2)^2}} = -2\int_{0}^{2}\frac{dv}{\sqrt{1+v^2}} = -2\sinh^{-1}(2)=e^{-2}-e^2.\]